Electric machines provide the conversion of mechanical energy to electrical energy (in the generating mode) and vice versa (in the motoring mode), via magnetic energy storage in their airgap (mainly) and in their permanent magnets (PMs) (if any). They are crucial in electric energy conversion and processing and for motion (torque, speed, and position) control in all industries, for better productivity, energy savings, and less harm to environment. Traveling field electric machines are still standard as synchronous and induction machines. The former (synchronous) need dc rotor excitation, or PMs or magnetically salient rotor to create a-fixed-to-rotor traveling field while the latter (induction) implies a cage (or wound) rotor winding. The average torque is nonzero as stator and rotor traveling fields are at standstill with each other. In an effort to take advantage of the variable reluctance concept in producing nonzero magnetic co-energy (and torque) in the machine versus variation of rotor position, better reluctance synchronous machines (RSMs) (line-start or inverter-fed) in terms of efficiency <inline-formula> <tex-math notation="LaTeX">$\times $ </tex-math></inline-formula> power factor, torque density, and costs without and with assisting PMs have been introduced lately. High magnetic saliency is implicit and requires a high ratio of pole pitch to airgap and a distributed ac stator winding, in the hope to meet the IEC 4, 5 standards of efficiency. On the other hand, better magnetic materials (with lower core losses) and higher fundamental frequency provided by pulsewidth modulation (PWM) static power converter (including silicon carbide (SiC) arrival) up to 1 kHz, in general, and the high price (and scarcity) of high energy PMs have triggered formidable Research and Development worldwide efforts to investigate many machine topologies without and with PMs (mainly ferrites or bonded NdFeB of lower cost), with mainly concentrated ac windings based on the principle of “flux-modulation (F-M)” provided by variable reluctance (magnetic anisotropy) concept. This article aims to summarize recent progress in reluctance electric machines, without and with PM assistance of synchronous and “F-M” types in terms of topologies, modeling, design, control, and merits and demerits based on comprehensive-technical and economic performance indexes.